Removing the aces from Deck A and putting them into Deck B transfers one heart and three other cards. The result is that:

12 out of 49 cards in Deck A will be hearts - this is  of the cards; and,

14 out of 57 cards in Deck B will be hearts - this is  of the cards.

To compare these probabilities, we express them in a common denominator as follows:

Deck A: 

Deck B: 

The probability that a random draw from Deck B will yield a heart is slightly higher.

Catherine has a bag of marbles with 5 red marbles, 6 green marbles and 7 blue marbles. What is the probability that Catherine will choose a red marble when she reaches into the bag? 

In order to find the probability of anything we must divide the part by the whole. 

We know that the part in this case are the red marbles; there are 5 of them. 

To find the whole, we must add all the parts: 

The number of balls in the box, in total, will be

 balls,

5 of which will be marked with a "5". 

The probability of drawing a "5" is therefore

The even numbers in the range of 1 to 10 are 2, 4, 6, 8, and 10; the number of balls marked with one of these numbers is 

Similarly, the number of balls marked with one of the odd numbers is

There are more even-numbered balls than odd, so the probability of drawing an even-numbered ball is the greater.

There are   possible rolls of two fair six-sided dice, each of which will come up with equal probability. The set of rolls is shown below, with the ways to roll an even number indicated.

There are 18 ways out of 36 to roll an even sum, making the probability of this event equal to .

There are   possible rolls of two fair six-sided dice, each of which will come up with equal probability. The set of rolls is shown below, with the ways to roll a 7 or less indicated.

There are 21 ways out of 36 to roll a sum of 7 or less, making the probability of this outcome .

Box 1 contains the odd-numbered balls, of which there are

.

One of these balls is marked with a "1", making the probability of drawing this ball .

Box 2 contains the even-numbered balls, of which there are

.

Two of these balls are marked with a "2", making the probability of drawing such a ball .

Between two fractions with the same numerator, the one with the lesser denominator is the greater, so , and the greater probability is that of drawing a "2" from Box 2.

Box 1 contains the odd-numbered balls, of which there are

.

Nine of these balls are marked with a "9", making the probability of drawing one of these balls .

Box 2 contains the even-numbered balls, of which there are

.

Ten of these balls are marked with a "10", making the probability of drawing such a ball .

We can compare these fractions by expressing both with a common denominator. , so we can rewrite these probabilities as 

and 

.

The probability that a ball randomly drawn from Box 1 will be marked "9" is the greater.

Each number will be represented by one red ball, so there will be ten red balls in the box.

The prime numbers - the numbers that have only 1 and themselves as factors - are 2, 3, 5, and 7, so there will be four green balls.

The composite numbers - the numbers that have more than two factors - are 4, 6, 8, 9, and 10, so there will be five blue balls. 

Note that 1 is neither prime nor composite.

Including the unmarked yellow ball, the total number of balls is , five of which are blue, so the probability of drawing a blue ball at random is .

One fourth of the cards in a standard deck of 52, not including the joker, will be diamonds; if two standard decks are shuffled together, then one fourth of those cards will be diamonds. The probability that Mary will draw a diamond is therefore .

However, when the joker is added to John's deck, there are more cards, but just as many diamonds. Therefore, the probability of John drawing a diamond must decrease to slightly less than . This gives Mary a greater probability of drawing a diamond.